A tutorial introduction to Bayesian inference for stochastic epidemic models using Approximate Bayesian Computation.

Likelihood-based inference for disease outbreak data can be very challenging due to the inherent dependence of the data and the fact that they are usually incomplete. In this paper we review recent Approximate Bayesian Computation (ABC) methods for the analysis of such data by fitting to them stochastic epidemic models without having to calculate the likelihood of the observed data. We consider both non-temporal and temporal-data and illustrate the methods with a number of examples featuring different models and datasets. In addition, we present extensions to existing algorithms which are easy to implement and provide an improvement to the existing methodology. Finally, R code to implement the algorithms presented in the paper is available on https://github.com/kypraios/epiABC.

[1]  G. Roberts,et al.  Bayesian inference for partially observed stochastic epidemics , 1999 .

[2]  P. O’Neill,et al.  Bayesian inference for stochastic epidemics in populations with random social structure , 2002 .

[3]  O. Cappé,et al.  Population Monte Carlo , 2004 .

[4]  W. Wheeler,et al.  Control of Communicable Diseases in Man. , 1961 .

[5]  Dennis Prangle,et al.  Summary Statistics in Approximate Bayesian Computation , 2015, 1512.05633.

[6]  Ellen Brooks-Pollock,et al.  A dynamic model of bovine tuberculosis spread and control in Great Britain , 2014, Nature.

[7]  D. Gillespie.  Exact Stochastic Simulation of Coupled Chemical Reactions , 1977 .

[8]  Richard J Boys,et al.  Bayesian inference for stochastic epidemic models with time-inhomogeneous removal rates , 2007, Journal of mathematical biology.

[9]  Theodoros Kypraios,et al.  Efficient Bayesian inference for partially observed stochastic epidemics and a new class of semi-parametric time series models , 2007 .

[10]  J. Fox,et al.  Viruses in families: surveillance of families as a key to epidemiology of virus infections. , 1980 .

[11]  T. Lindvall.  Lectures on the Coupling Method , 1992 .

[12]  M. Blum.  Approximate Bayesian Computation: A Nonparametric Perspective , 2009, 0904.0635.

[13]  Mark M. Tanaka,et al.  Sequential Monte Carlo without likelihoods , 2007, Proceedings of the National Academy of Sciences.

[14]  David Welch,et al.  Approximate Bayesian computation scheme for parameter inference and model selection in dynamical systems , 2009, Journal of The Royal Society Interface.

[15]  Alexander Grey,et al.  The Mathematical Theory of Infectious Diseases and Its Applications , 1977 .

[16]  George Streftaris,et al.  Bayesian analysis of experimental epidemics of foot–and–mouth disease , 2004, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[17]  S. Sisson,et al.  A comparative review of dimension reduction methods in approximate Bayesian computation , 2012, 1202.3819.

[18]  M. Feldman,et al.  Population growth of human Y chromosomes: a study of Y chromosome microsatellites. , 1999, Molecular biology and evolution.

[19]  G. Roberts,et al.  A novel approach to real-time risk prediction for emerging infectious diseases: a case study in Avian Influenza H5N1. , 2009, Preventive veterinary medicine.

[20]  N. Ferguson,et al.  Transmission Parameters of the 2001 Foot and Mouth Epidemic in Great Britain , 2007, PloS one.

[21]  Martin Eichner,et al.  Transmission potential of smallpox: estimates based on detailed data from an outbreak. , 2003, American journal of epidemiology.

[22]  Yanan Fan,et al.  Likelihood-Free MCMC , 2011 .

[23]  F. Ball,et al.  Epidemics with two levels of mixing , 1997 .

[24]  A N Pettitt,et al.  Bayesian modelling of an epidemic of severe acute respiratory syndrome , 2006, Bulletin of mathematical biology.

[25]  Damian Clancy,et al.  Exact Bayesian Inference and Model Selection for Stochastic Models of Epidemics Among a Community of Households , 2007 .

[26]  C. Fraser,et al.  Household transmission of 2009 pandemic influenza A (H1N1) virus in the United States. , 2009, The New England journal of medicine.

[27]  G. Roberts,et al.  Bayesian analysis for emerging infectious diseases , 2009 .

[28]  Arnaud Doucet,et al.  An adaptive sequential Monte Carlo method for approximate Bayesian computation , 2011, Statistics and Computing.

[29]  T Déirdre Hollingsworth,et al.  Controlling infectious disease outbreaks: Lessons from mathematical modelling , 2009, Journal of public health policy.

[30]  Aaron Smith,et al.  The use of a single pseudo-sample in approximate Bayesian computation , 2014, Stat. Comput..

[31]  A. Cook,et al.  Inference in Epidemic Models without Likelihoods , 2009 .

[32]  D. Balding,et al.  Analyses of infectious disease data from household outbreaks by Markov chain Monte Carlo methods , 2000 .

[33]  I M Longini,et al.  Estimating household and community transmission parameters for influenza. , 1982, American journal of epidemiology.

[34]  C C Drovandi,et al.  Estimation of Parameters for Macroparasite Population Evolution Using Approximate Bayesian Computation , 2011, Biometrics.

[35]  D. Balding,et al.  Approximate Bayesian computation in population genetics. , 2002, Genetics.

[36]  Philip D. O'Neill,et al.  Computation of final outcome probabilities for the generalised stochastic epidemic , 2006, Stat. Comput..

[37]  M. J. Bayarri,et al.  Non-Centered Parameterisations for Hierarchical Models and Data Augmentation , 2003 .

[38]  Dennis Prangle,et al.  Adapting the ABC distance function , 2015, 1507.00874.

[39]  Gavin J. Gibson,et al.  Estimating parameters in stochastic compartmental models using Markov chain methods , 1998 .

[40]  Stuart Barber,et al.  The Rate of Convergence for Approximate Bayesian Computation , 2013, 1311.2038.

[41]  Janina Muller,et al.  Analysis Of Infectious Disease Data , 2016 .

[42]  A. Pettitt,et al.  Bayesian inference of hospital-acquired infectious diseases and control measures given imperfect surveillance data. , 2007, Biostatistics.

[43]  PETER NEAL,et al.  A case study in non-centering for data augmentation: Stochastic epidemics , 2005, Stat. Comput..

[44]  P. Moral,et al.  Sequential Monte Carlo samplers , 2002, cond-mat/0212648.

[45]  Susan S. Huang,et al.  Assessing the role of undetected colonization and isolation precautions in reducing Methicillin-Resistant Staphylococcus aureus transmission in intensive care units , 2010, BMC infectious diseases.

[46]  C. Viboud,et al.  A Bayesian MCMC approach to study transmission of influenza: application to household longitudinal data , 2004, Statistics in medicine.

[47]  Paul Marjoram,et al.  Markov chain Monte Carlo without likelihoods , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[48]  Gareth O. Roberts,et al.  Non-centred parameterisations for hierarchical models and data augmentation. , 2003 .

[49]  Thomas Sellke,et al.  On the asymptotic distribution of the size of a stochastic epidemic , 1983, Journal of Applied Probability.

[50]  Guosheng Yin,et al.  Bayesian generalized method of moments , 2009 .

[51]  J S Noel,et al.  A Viral Gastroenteritis Outbreak Associated with Person-to-Person Spread Among Hospital Staff , 1998, Infection Control & Hospital Epidemiology.

[52]  Arnaud Guyader,et al.  New insights into Approximate Bayesian Computation , 2012, 1207.6461.

[53]  Peter Neal,et al.  Efficient likelihood-free Bayesian Computation for household epidemics , 2012, Stat. Comput..